I found half of a solution -- it's not perfect but it's better than nothing. If anyone else has other ideas, _please_ let me know =).
Settings > Compiler and Debugger > Other (tab) > Advanced > Others (tab) > Use Flat Objects
This puts all of the *.o files into the top level directory. Not perfect, but it's better than nothing =).
However, this still does not fix the problem. I still get
/home/muppetjones/Ubuntu One/projects/test/main.cpp|9|undefined reference to `bStat::bStat()'|
Even though it finds the *.h file which is in the same directory as the *.o file and the *.cpp!!
What am I missing?!